In this work, we seek to combine an accurate and flexible discontinuous DG space discretization with an implicit-explicit (IMEX) time discretization, see e.g. [1]}, [2]}, to obtain an efficient method for compressible flow of real gases at low to moderate Mach numbers. Our goal is to derive a method that can then be easily extended to handle multiphase compressible flows, where a number of coupling and forcing terms arise that cannot be dealt with efficiently by straightforward application of conventional solvers.
In order to obtain a method that is robust in the low Mach number limit, following [3]}, [4]}, we couple implicitly the energy equation to the momentum equation, while treating the continuity equation in an explicit fashion.
Notice that a conceptually similar approach has been used in [5]}, [6]} for the discretization
employed in the IFS-FVM atmospheric model.
In order to obtain a formulation that is efficient also in presence of non negligible viscous terms, we resort to an operator splitting approach, see e.g. [7]}.
More specifically, as commonly done in numerical models for atmospheric physics, we split the hyperbolic part of the problem, which is treated by an IMEX extension of the method proposed in [4]}, from the diffusive terms, which are treated implicitly. Second order accuracy can then be obtained by the Strang splitting approach [7]}, [10]}.
Notice that, with respect to the IMEX approach proposed for the Euler equations in [11]}, the technique presented here does not require to introduce reference solutions, does not introduce inconsistencies in the splitting with respect to a reference solution and
only requires the solution of linear system of a size equal to that of the number of discrete degrees of freedom needed to describe a scalar variable, as in [4]}.
